use matrix-free GMRES to solve
the Newton update problems with implicit sensitivity calculation,
i.e., the steps enclosed by the double dashed block
-in Fig.~\ref{fig:ef_flow}.
+in Figure~\ref{fig:ef_flow}.
Then implementation issues of GPU acceleration
will be discussed in detail.
Finally, the Gear-2 integration is briefly introduced.
%% \end{algorithm}
\begin{algorithm}
-\caption{Standard GMRES\index{iterative method!GMRES} algorithm.} \label{alg:GMRES}
+\caption{standard GMRES\index{iterative method!GMRES} algorithm} \label{alg:GMRES}
\KwIn{ $ A \in \mathbb{R}^{N \times N}$, $b \in \mathbb{R}^N$,
and initial guess $x_0 \in \mathbb{R}^N$}
\KwOut{ $x \in \mathbb{R}^N$: $\| b - A x\|_2 < tol$}
the small size of Hessenberg matrix,
and the frequent inspection of values by host, it is
preferable to allocate $\tilde{H}$ in CPU (host) memory.
-As shown in Fig.~\ref{fig:gmres}, the memory copy from device to host
+As shown in Figure~\ref{fig:gmres}, the memory copy from device to host
is called each time when Arnoldi iteration generates a new vector
and the orthogonalization produces the vector $h$.
